Bosonic entanglement renormalization circuits from wavelet theory

We would like to thank the reviewer for their careful assessment of our work and their suggestions for improving the manuscript.

Regarding the comment that the presentation might possibly be too succinct and assumes some familiarity with previous works on wavelets and entanglement renormalization: apart from adding some more helpful explicit calculations (see below), we have also added Appendix A, which reviews the construction of these previous works in the fermionic setting. This appendix also includes Table 1, which contrasts the bosonic and the fermionic setting to make the similarities and differences more clear. While these previous works are in the fermionic setting and we build our story "from the ground" for the bosonic setting, we agree that this appendix should be rather beneficial to the reader.

We now list the changes suggested by the reviewer and how we implemented them:

• An obvious observation is that the authors could have provided the explicit calculations in the appendices, so to save the reader to have to sit and reproduce them. For example, the calculation of the reconstruction from scaling and wavelets.

Reply: We have added explicit calculations at various locations, especially in Section 5. The decomposition and reconstruction formulas are given on page 5 (in the discrete case) and on page 11 (in the continuous case).

• Regarding the continuum limit in the massless case, the interesting analytical result is that the number of descendants depends on the number of vanishing moments of the wavelet filters. How does this concept maps to the more accepted idea that the larger the bond dimension the larger the number of correct descendent becomes? I understand that vanishing moments implies that the wavelet is orthogonal in L² to polynomials, but what does this say to its non-vanishing support, that I would expect to be related to the bond dimension? If this point is discussed I have missed the discussion.

Reply: That is a good question and the answer was indeed rather implicit in our discussion. It is true that K vanishing moments (and hence K descendants) requires a filter size of at least 2K. In fact, in our explicit construction the filter size is 2K + 4L where L controls the accuracy of the approximation of the dispersion relation. We have added a sentence at the end of Section 5.2 to explain this better. Regarding the bond dimension: While in the fermionic setting the size of the filter is directly related to the bond dimension, in the bosonic setting the bond dimension is formally infinite. However, the size of the filter is (in both cases) still directly related to the circuit depth of a single layer (see the first sentence of Section 4; the construction is given in Appendix C).

• Is there any relation of the present construction with the magic ER, and the cMPO appearing there acting as disentangler? Can the cMPO mentioned there be constructed from the wavelets? In the section about the relation to finite depth Gaussian circuit the authors seem to suggest that the construction is general, I wonder if they can find the explicit one for the magic ER. There also some of the descendent could be computed analytically if I am not wrong. Can they comment about this?

Reply: That is an interesting question. We have thought quite a bit about the relation to cMERA in general, and there are definitely some strong similarities. In particular, there is a similarity between cMERA and the continuous wavelet transform (CWT). It would be very pleasing to find way to 'discretize' a cMERA to a MERA, analogously to how it is known that a CWT can be discretized to discrete wavelet transform provided the wavelet function used in the CWT satisfies some particular conditions (the wavelet functions allowed in the CWT are a much broader class than those we consider, which arise from a filter). However, at least to our best current understanding, there are some subtleties in making this precise and the correspondence does not quite work out yet. We discuss this briefly in Section 5.4. We think that this would be very interesting to explore in more depth in future work.

• One of the basic features of CFTs is that the dispersion relation is linear. Equation 4 seems to suggest that the renormalized dispersion relation becomes quadratic, is this the case? Or does it become linear + quadratic thus extrapolating from linear al low k to quadratic at high k?

Reply: The dispersion relation of the continuum limit is indeed linear. Indeed, one can show that if the filters are related as in (4) with the dispersion relation of the massless harmonic chain, then the corresponding wavelet functions are related by a linear dispersion relation. Previously this was mentioned without calculation in Section 5.2. We have now added a complete derivation of this fact (see equation (19) and above).

• The authors could have definitely done a better job at including the relevant material to facilitate the reproduction of the main results and at discussing the relation of their results to the more widely known ones.

Reply: As outlined above, we have significantly revised the manuscript by expanding the introduction, elaborating technical calculations (which in particular should make it easier to verify and reproduce our main results), adding supporting numerical results, and adding an appendix that summarizes prior work in the fermionic setting and compares it with our bosonic results.

We would like to thank the reviewer for their kind words. The question about building quantum circuits for interacting theories is very relevant. Unfortunately we have at this point no results in this direction to share. We added a remark that a perturbation theory approach has been investigated for cMERA, also around a Gaussian ansatz, which may serve as inspiration. Hopefully this will be understood better in future research!

We would like to thank the reviewer once more for their thoughtful review of our work. We have tried our best to implement the suggested changes, which we believe has already considerably improved the readability of the revised manuscript.

Regarding the general relevance of our work, we commented in our previous author reply on some of the key innovations of our work: (1) It is to our knowledge the first wavelet-based construction an entanglement renormalization scheme approximating the ground state of a bosonic system, resolving an open problem mentioned in Phys. Rev. Lett. 116, 140403 (the paper that pioneered the use of wavelets in the fermionic setting). (2) We provide a solution for general free bosonic models. In contrast, in the fermionic setting essentially only a single model is understood from the wavelet perspective. Together, we believe that our work provides a significant advance in our understanding of the wavelet-MERA correspondence. We have revised the introduction of the paper to explain these points better. We have also added Table 1 (in appendix A) that makes an explicit comparison with the fermionic wavelet-MERA correspondence.

We now list the changes suggested by the reviewer and how we implemented them:

1- The authors mention Tensor Networks in the abstract and renormalisation methods in the introduction. At this stage, I think it would be beneficial for the reader to properly introduce Tensor Networks providing some more insights and how they connect to quantum circuits together with more references especially as there is a rich literature available e.g. the following for an overview: R. Orús, Nature Reviews Physics 1, 538 (2019); S. Montangero, Introduction to Tensor Network Methods (Springer International Publishing, 2018); R. Orús, Nature Reviews Physics 1, 538 (2019); J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics 82, 277306 (2010); Or the following for MERA and the connection to quantum circuits: Phys. Rev. Lett. 101, 110501; Phys. Rev. X 10, 041038; William Huggins et al 2019 Quantum Sci. Technol. 4 024001;

Reply: We have expanded the introduction accordingly and now cite these relevant references.

2- In the introduction, for the sake of giving the reader a clear overview, I would appreciate a brief introduction into the different topics the manuscript builds on (wavelet theory, quantum circuits, entanglement renormalisation) with more cross-references and a more detailed description on why they are scientifically important, before describing how the work is to be categorized in the fields. E.g. I would appreciate many of the valuable insights given in section 5.4 already in the introduction which would provide a more comprehensive overview for the reader.

Reply: We have added some more context and now give brief introductions to the different topics that the manuscript builds on. We have moved part of the discussion in section 5.4 to the introduction. To improve readibility, we now separate this background material from the summary of our main results (new Subsection 1.1), and we have added a new Subsection 1.2 that outlines the organization of the paper.

3- The authors provide some numerical analysis in the appendix, however, I am missing some more numerical analysis as evidence for the correctness of the proposed discretisation and renormalisation scheme. In particular, I would suggest an analysis in which the proposed MERA-like structure is implemented and benchmarked against known ground state preparation results or against exactly calculatable ground states. (E.g. Showing the relative errors in the energy density for the approximate ground states prepared by the circuits, or showing two-point correlation functions in the approximated ground states)

Reply: We added Figure 4, which shows two-point correlation functions as produced by our construction for the massless harmonic chain (and different parameters K, L) with those of the exact ground state. This numerically illustrates our theorem (which bounds the approximation error of two-point functions). This figure had unfortunately been removed from an earlier version of the manuscript before submission to SciPost.

4- Technically, the tensor network – or quantum circuit structure – would be a dMERA ( arXiv:1711.07500) and not a typical MERA. This might be worth mentioning.

Reply: We now mention this explicitly in the introdution.

5- On page 4, I think it would help to define W_g properly before using it in the signal decomposition.

Reply: We have now made the definition more clear by first defining f^low and f^high and then W_g.

6- On page 5, the authors use the variable l (\ell) for the second time. However, it was first used as an index in the sum on page 4, while on this occasion it stands for the frequency mode (I guess). It would be helpful to properly define the variable l (\ell) here.

Reply: Thank you for pointing this out. The symbol l on page 5 was indeed not yet defined; we have replaced it by "number of layers".

7- On page 8, the authors change the notation from “the l-th layer” to “the \mathcal{L}-th layer” in and after the theorem.

Reply: This was intentional, but indeed confusing in the previous text: "l" is an index for an arbitrary circuit layer, while \mathcal{L} denotes the total number of renormalization layers in the approximation. The text has been changed to explain this better.

8- On page 8 at the beginning of section 5, the authors write “It turns out the scaling functions are a natural UV cut-off that is compatible with the entanglement renormalisation circuits, and in the critical case we find that we can reproduce certain conformal data exactly from a single layer of renormalisation.”. However, I find this statement as a non-trivial step for investigating the continuum limit and the authors do not provide any mathematical or numerical evidence here, nor any source supporting the claim.

Reply: These claims are discussed in Section 5.2 and 5.3. As our previous discussion was rather concise, we have now expanded Section 5.2 significantly to explain this in more detail.

9- On page 12, I am a bit puzzled, as the authors write “This poses some interesting questions” and later on “The wavelet filter design problem also leaves some open questions” but afterwards the authors do not pose a question or discuss the open problems in detail.

Reply: This was admittedly worded rather confusingly. We have now revised this part of the conclusions to make clear what precisely the open problems are.

10- On page 14, I would suggest enumerating the figures and referring to the indications in the caption for the sake of clarity.

Reply: We have implemented this change.

We would like to thank the reviewer for the extensive and helpful comments! As soon as the second review comes in we will be very happy to implement the suggested changes. Before that, we would like to state more clearly the innovation in our work. To our best knowledge, the connection between entanglement renormalization of free bosonic models and wavelets has not been worked out before. The reviewer mentions two references, New J. Phys. 12 025007 and Phys. Rev. X 8, 011003. Our work can indeed be seen as a synthesis between these two earlier works. However, in New J. Phys. 12 025007 there is no mention of wavelet theory, whereas Phys. Rev. X 8, 011003 is purely discussing the fermionic case. We are under the impression that our work is the first wavelet based construction an entanglement renormalization scheme approximating the ground state of a bosonic system. In fact, the original work on the connection between wavelets and MERA, Phys. Rev. Lett. 116, 140403, states this as an open problem in its discussion, and we are not aware of any later work providing a solution.
We would also like to mention that the current understanding of the fermionic case is restricted to a single critical model (of hopping fermions) and it is for instance not clear how to add a mass term. In contrast, our bosonic formalism works for general free bosonic models (whether critical or not), and it also comes with an algorithm that constructs appropriate wavelets for a general dispersion relation.
In our view this provides a significant extension of the wavelet-MERA correspondence on a level that is appropriate for SciPost Physics.

Finally, we see this work as mainly a theoretical one. In a previous version (https://arxiv.org/abs/2004.11952v1) we had a figure illustrating our theorem with some numerics (figure 4, for the critical case), but felt it perhaps didn't add much to the theorem. Of course the reviewer makes a good point that this is a little unsatisfying and we are happy to restore it!
In the updated version we will also try to explain more carefully what precisely is new in our work, as this is indeed not very clear from our discussion of the existing literature.